Introduction to Square Root
Any number can be expressed as the product of the prime numbers. This method of representation of a number in terms of the product of prime numbers is termed as the prime factorization method. It is the easiest method known for the manual calculation of the square root of decimal numbers. But this method becomes tedious and tiresome when the amount involved is large. In order to beat this problem, we use the division method.
Consider the following method for finding the square root of the decimal number. It is explained with the assistance of an example for a transparent understanding.
Note: The number of digits in a perfect square is very significant for calculating its square root of a decimal number by the long division method.
What is Square Root of Decimal Number
The square root of decimals is calculated in the same way as the square root of whole numbers.
Inverse operations include taking the square root of a number and squaring a number. The square root of a number is the number that is multiplied by itself to give the original number, whereas the square of a number is the value of the number's power 2.
The value of a decimal number raised to the power 1/2 is called the square root of the decimal. The square root of 24.01, for example, is 4.9, as (4.9)2 = 24.01.
The estimation approach or the long division method can be used to calculate the square root of a decimal value.
The pairings of whole number parts and fractional parts are separated using bars in the long division method.
After that, long division is performed in the same manner as any other whole number.
Steps for Finding the Square Root of Decimal Number with Examples
Square Root: Estimation Method
Estimation and approximation are ways to make calculations easier and more realistic by making a good guess of the real value.
This method can also help you figure out and approximate the square root of a number you're given.
We only need to find the perfect square numbers that are closest to the given decimal number to figure out its approximate square root value.
Let's find the square root of 31.36. Below are the steps:
Find the perfect square numbers that are closest to 31.36.
The perfect square numbers closest to 31.36 are 25 and 36.
√25 = 5 and √36 = 6. This means that √31.36 is somewhere between 5 and 6.
Now, we must determine if √31.36 is closer to 5 or 6.
Let’s consider 5.5 and 6.
5.52 = 30.25 and 62= 36. As a result, √31.36 is near to 5.5 and lies between 5.5 and 6.
As a result, the square root of 31.36 is around 5.5.
Square Root: Long Division Method
When we need to divide big numbers into steps or parts, we utilize the long division approach, which breaks the division problem down into a series of simpler steps.
Using this strategy, we may get the precise square root of any number.
Let's find the square root of 2.56. Below are the steps:
Place a bar over each pair of digits starting with the unit. We will have two pairs, i.e. 2 and 56.
Then divide it by the biggest number whose square is less than or equal to it.
Here, the whole number part is 2 and we have 1 x 1 = 1. So, the quotient is 1.
Reducing the number, that is, the pair of fractions under the remainder bar (that is 1).
Add the quotient's last digit to the divisor, which is 1 + 1 = 2. Find a suitable number to the right of the obtained sum (that is 2) that, when combined with the sum's result, provides a new divisor for the new dividend (that is, 156) that is brought down. As we get closer to the fractional part, add a decimal after 1 in the quotient.
The new quotient number will be the same as the divisor number, therefore, the divisor will now be 26 and the quotient will be 1.6 because 26 x 6 = 156. (The criterion is the same — the dividend must be less than or equal to it.)
Using a decimal point, continue the process by adding zeros in pairs to the remainder.
The resultant quotient is the square root of the number. As a result, 2.56 has a square root of 1.6.
Examples to be Solved
Question 1: Find the square root of decimal number 29.16.
Solution: The following steps will explain how to find the square root of decimal number 29.16 by using the long division method:
Step 1. Write down the decimal number and then make pairs of the integer and fractional parts separately. Then, the pair of the integers of a decimal number is created from right to left and so, the pair of the fractional part is made right from the start of the decimal point.
Example: Within the decimal number 29.16, 29 is one pair and then 16 is another pair.
Step 2. Find the amount whose pair is a smaller amount than or adequate to the primary pair. In the number 29.16, 5’s square is adequate to 25. Hence, we'll write 5 within the divisor and 5 within the quotient.
Step 3. Now, we will subtract 25 from 29. The answer is 4. We will bring down the opposite pair which is 16 and put the percentage point within the quotient.
Step 4. Now, we'll multiply the divisor by 2. Since 5 into 2 is adequate to 10, so we'll write 10 below the divisor. We need to seek out the third digit of the amount in order that it's completely divisible by the amount 416. We already have two digits 10. The 3rd digit should be 4 because 104 . 4 = 116.
Step 5. Write 4 in the quotient's place. Hence, the answer is 5.4.
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Question 2: Find the square root of decimal number 84.64 by using a long division method.
Solution: Follow these steps to seek out the root of this decimal number 84.64.
Step 1. Write down the decimal number and then make pairs of the integer and fractional parts separately. The pair of the integers of a decimal number is made from right to left and so the pair of the fractional part is made right from the start of the decimal point.
So, in the decimal number 84.64, 84 is one pair and then 64 is another one.
Step 2. Find the amount whose pair is a smaller amount than or adequate to the primary pair. In the number 84.64, 9’s square is equal to 81. Hence, we'll write 9 within the divisor and 9 within the quotient.
Step 3. Now, we'll subtract 81 from 84. The answer is 3. We will bring down the opposite pair which is 64 and put the percentage point within the quotient after 9.
Step 4. Now, we'll multiply the divisor by 2. Since 9 into 2 is adequate to 18, so we'll write 18 below the divisor. We have to find out the third digit for the number so that it is totally divisible by the number 364. We already have two digits 18. The 3rd digit should be 2 because 182 . 2 = 364.
Step 5. Write 2 within the quotient's place after the percentage point. Hence, the answer is 9.2.
FAQs on Square Root of Decimal Number
1.What are the methods to find the square root of a number?
There are four ways of determining the square root of a number. They are as follows:
Repeated subtraction method of the square root
The consecutive odd numbers are to be subtracted from the number for which the square root is to be found, till 0 is achieved. The square root of the given integer is the number of times we subtract.
Square root by prime factorization method
The representation of a number as a product of prime numbers is known as "prime factorization."
Square root by estimation method
To make calculations easier and more realistic, estimation and approximation refer to a plausible assumption of the actual number.
Square root by long division method
Long division is a technique for breaking down large numbers into smaller pieces or sections by breaking down the operation into a series of smaller phases.
2.When am I going to have to multiply square roots?
Students always raise that question, regardless of how thoroughly they are being taught the algebraic ideas.
Multiplying square roots, for example, is necessary for:
Architects
Artists
Carpenters
Construction workers
Designers
Engineers
Hence, it is clear that those professionals have studied math in school and continue to apply the concept in their profession.
While some students will be required to solve equations daily, others will use the principles to create estimates.
3.How do we multiply decimals?
When it comes to grouping items, multiplication of decimals is crucial. Multiplying decimals is similar to multiplying whole integers, with the exception that the decimal point is placed in the product.
When multiplying decimals, ignore the decimal point and multiply the numbers; the total number of decimal places in both values equals the number of decimal places in the product.
If the product's number of decimal places exceeds the number of digits, zeros can be appended to the left before the decimal point is inserted.
4.What are the application of Square roots?
Square roots are used extensively in mathematics and have applications in a wide range of fields, including probability, statistics, physics, architecture, and engineering.
Here are some examples of how square roots are used in everyday life:
Finance (to calculate the rate of return on an asset over a two-unit period – for example, 2 years, 2 months, etc.)
Normal distributions (to define the function that gives us a normal distribution curve and the probability density function)
Pythagorean Theorem (to find the lengths and distances of triangle sides in two dimensions or 3 dimensions)
Quadratic formula (if we want to know when a falling object reaches a given height)
The radius of circles (if we want to know the radius of a circle with a given area)
Simple Harmonic Motion (for Pendulums and Springs)
Standard deviation (measuring the spread of data).
5.What is a decimal number?
A decimal number is defined in algebra as a number that contains a decimal point dividing the whole number from its fractional portions. A decimal point is denoted by a dot. A value less than one is represented by the digits after the decimal point.
To calculate decimals, the preceding powers of ten are used. As we progress from left to right, the place value of digits is divided by 10, resulting in tenths, hundredths, and thousandths being determined by the decimal place value.
